Question:
In any ΔABC, prove that
$a \sin A-b \sin B=c \sin (A-B)$
Solution:
Need to prove: $a \sin A-b \sin B=c \sin (A-B)$
Left hand side,
$=a \sin A-b \sin B$
$=(b \cos C+c \cos B) \sin A-(c \cos A+a \cos C) \sin B$
$=b \cos C \sin A+c \cos B \sin A-c \cos A \sin B-a \cos C \sin B$
$=c(\sin A \cos B-\cos A \sin B)+\cos C(b \sin A-a \sin B)$
We know that, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$ where R is the circumradius.
Therefore,
$=c(\sin A \cos B-\cos A \sin B)+\cos C(2 R \sin B \sin A-2 R \sin A \sin B)$
$=c(\sin \mathrm{A} \cos \mathrm{B}-\cos \mathrm{A} \sin \mathrm{B})$
$=c \sin (A-B)$
$=$ Right hand side. [Proved]
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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